Question: How to find the algebraic equation of the intersection curve(s) of two cylinders?


I'm stucked in determining the intersection curve(s) of two (intersecting !) cylinders.
Plotting these curves can easily be done with plots:-intersectionplot, but I'm interested in finding the algebraic equations of this (these) curve(s).

I tried to do this while using either parametric or implicit representations of the two cylinders.

(For now on I'm using Maple 2015 and I wasn't capable to repoduce a few promising results I'd obtained at the office with Maple 2019 and parametric representations. So I mainly concentrated onimplicit representations).

If E(x,y,z) and E'(x,y,z) denote implicit representations of cylinders C and C', I had (naively) thought that simply solving 
E(x,y,z) = E'(x,y,z) with respect with x, y and z would have done the job.

Unfortunately, even for the simplest case of orthogonal circular cylinders of same radii, solve returns the couple of planes which contain the two intersection curves (ellipses) but not these ellipses themselves

Maybe there is a "simple" way to obtain the algebraic equation(s) of the intersection curve(s) but I wasn't capable to find it.
Instead of that I wrote a complicated stuff (please look to the attached file) which works well in some situations and not in others (see the last test case).

Could you please help me to answer this issue?

Thanks in advance

PS: no real need to consider tangent cylinders along a generatrix or one-point tangency.

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